Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method

Maria Rosaria Formica, Eugeny Ostrovsky, Leonid Sirota

Research output: Contribution to journalArticlepeer-review

Abstract

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi.

Original languageEnglish
Pages (from-to)399-413
Number of pages15
JournalActa Applicandae Mathematicae
Volume170
Issue number1
DOIs
StatePublished - 1 Dec 2020

Bibliographical note

Funding Information:
The first author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project “sostegno alla Ricerca individuale”.

Funding Information:
The first author has been partially supported by the Gruppo Nazionale per l?Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Universit? degli Studi di Napoli Parthenope through the project ?sostegno alla Ricerca individuale?.

Publisher Copyright:
© 2020, Springer Nature B.V.

Keywords

  • Chapman-Kolmogorov equation
  • Fundamental solution
  • Generalized Mittag-Leffler function
  • Neumann series
  • Partial Differential Equation of parabolic type
  • Volterra’s integral equation

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